Cook the pasta: Bring a large pot of salted water to a rolling boil. Add pasta and cook until just al dente.
Drain and rinse briefly under cool water to stop the cooking. Shake off excess water well.
Whisk the dressing: In a bowl or jar, whisk together olive oil, red wine vinegar, lemon juice, Dijon, garlic, oregano, honey, salt, pepper, and red pepper flakes if using. Taste and adjust acidity and salt.
Toss the warm pasta with dressing: Add the drained pasta to a large mixing bowl.
While still slightly warm, pour over about two-thirds of the dressing and toss. This helps the pasta absorb flavor.
Add chicken and veggies: Fold in chicken, tomatoes, cucumber, red onion, roasted red peppers, and olives. Gently toss to distribute evenly.
Add mozzarella and herbs: Add mozzarella, basil, and parsley.
Drizzle in the remaining dressing and toss lightly so the mozzarella stays intact.
Taste and finish: Season with more salt, pepper, or lemon if needed. Garnish with extra basil. Serve right away or chill for 30–60 minutes to let flavors meld.